1. P. Jin, Z. Zhang, A. Zhu, Y. Tang, G. E. Karniadakis, SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems, Natural Networks, Volume 132https://doi.org/10.1016/j.neunet.2020.08.017, 2020.
  2. G. Pang, M. D’Elia, M. Parks, G. E. Karniadakis, nPINNs: Nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications, Journal of Computational Physics Volume 422, December 2020.
  3. K. Shukla, P. C. Di Leoni, J. Blackshire, D. Sparkman, G. E. Karniadakis, Physics-Informed Neural Network for Ultrasound Nondestructive Quantification of Surface Breaking Cracks, Journal of Nondestructive Evaluation, Article number: 61, August 2020.
  4. A. D. Jagtap, K. Kawaguchi, G. E. Karniadakis, Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks, Proceedings of the Royal Society, July 2020.
  5. X. Menga, Z. Li, D. Zhang, G. E. Karniadakis, PPINN: Parareal physics-informed neural network for time-dependent PDEs, Computer Methods in Applied Mechanics and Engineering, Volume 370,  October 2020
  6. Y. Chen, L. Lu, G. E. Karniadakis, and L. D. Negro, Physics-informed neural networks for inverse problems in nano-optics and metamaterials, Optics Express, Vol. 28, Issue 8, pp. 11618-11633, 2020.
  7. Q. Zheng, L. Zeng, G. E. Karniadakis, Physics-informed semantic inpainting: Application to geostatistical modeling, Journal of Computational Physics, June 2020.
  8. P.Jin, L. Lu, Y. Tanga, G. E. Karniadakis, Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness, Neural Networks, July 2020.
  9. Q Zheng, L Zeng, Z Cao, GE Karniadakis, Physics-informed semantic inpainting: Application to geostatistical modeling, Journal of Computational Physics, 109676, 2020.
  10. AD Jagtap, E Kharazmi, GE Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws:   Applications to forward and inverse problems, Computer Methods in Applied Mechanics and Engineering 365, 113028, 2020.
  11. L Lu, M Dao, P Kumar, U Ramamurty, GE Karniadakis, S Suresh, Extraction of mechanical properties of materials through deep learning from instrumented indentation, Proceedings of the National Academy of Sciences, March 16, 2020.
  12. Z Mao, AD Jagtap, GE Karniadakis, Physics-informed neural networks for high-speed flows, Computer Methods in Applied Mechanics and Engineering 360, 112789, 2020.
  13. AD Jagtap, K Kawaguchi, GE Karniadakis, Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, Journal of Computational Physics 404, 109136, 2020.
  14. M Raissi, A Yazdani, GE Karniadakis, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367 (6481), 1026-1030, 2020.
  15. GCY Peng, M Alber, AB Tepole, WR Cannon, S De, .., GE Karniadakis, E. Kuhl, Multiscale Modeling Meets Machine Learning: What Can We Learn?, Archives of Computational Methods in Engineering, 1-2, 2020.
  16. X Meng, GE Karniadakis, A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems, Journal of Computational Physics 401, 109020, 2020.
  17. L Yang, D Zhang, GE Karniadakis, Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations, SIAM Journal on Scientific Computing 42 (1), A292-A317, 2020.
  18. D Zhang, L Guo, GE Karniadakis, Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks, SIAM Journal on Scientific Computing 42 (2), A639-A665, 2020.
  19. PP Mehta, G Pang, F Song, GE Karniadakis, Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network, Fractional Calculus and Applied Analysis 22 (6), 1675-1688, 2019.
  20. Z Mao, Z Li, GE Karniadakis, Nonlocal flocking dynamics: Learning the fractional order of PDEs from particle simulations, Communications on Applied Mathematics and Computation 1 (4), 597-619, 2019.
  21. D Fan, G Jodin, TR Consi, L Bonfiglio, Y Ma, LR Keyes, GE Karniadakis, …,MS Triantafyllou, A robotic Intelligent Towing Tank for learning complex fluid-structure dynamics, Science Robotics 4 (36), 2019.
  22. M. Alber, A. B. Tepole, W. R. Cannon, S. De, S. Dura-Bernal, K. Garikipati, G. Karniadakis, W. W. Lytton, P. Perdikaris, L. Petzold, E. Kuhl, “Integrating machine learning and multiscale modeling— perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences.” Digital Medicine 2:115, 2019.
  23. M. Raissi, P. Perikaris, G.E. Karniadakis, “Physics-informed neutral networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics 378, 686-707, 2019.
  24. D. Zhang, L. Lu, L. Guo, G.E. Karniadakis, “Quantifying total uncertainty in physics-informed neutral networks for solving forward and inverse stochastic problems”, Journal of Computational Physics, 2019.
  25. G. Pang, L. Lu, G.E. Karniadakis, “fPINNS: Fractional physics-informed neutral networks”, SIAM Journal on Scientific Computing 41 (4), A2603-A2626, 2019.
  26. M. Raissi, H. Babee, G.E. Karniadakis, “Parametric Gaussian process regression for big data”, Computational Mechanics 64 (2), 409-41, 2019.
  27. M. Gulian, M.Raissi, P. Perdikaris, G.E. Karniadakis, “Machine learning of space-fractional differential equations”, SIAM Journal on Scientific Computing 14 (4), A2485-A2509, 2019.
  28. A.L. Blumers, Z. Li, G.E. Karniadakis, “Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics.Journal of Comp. Physics 393, 214-228, 2019
  29. G. Pang, L Yang, G.E. Karniadakis, “Neural-net-induced Gaussian process regression for function approximation and PDE solution.Journal of Computational Physics 384, 270-288, 2019.
  30. S. Lee, F. Dietrich, G.E Karniadakis, I.G. Kevrekidis, “Linking Gaussian process regression with data-driven manifold embeddings for nonlinear data fusion.Interface focus 9 (3), 20180083, 2019.
  31. M. Rassi, Z. Wang, M.S. Triantafyllou, G.E. Karniadakis, “Deep Learning of vortex-induced vibrations.” Journal of Fluid Mechanics 861, 119-137, 2019.
  32. Z. Wang, , M. Triantafyllou,  Y. Constantinides, G. E. Karniadakis,  “An entropy-viscosity large eddy simulation study of turbulent flow in a flexible pipe.” J. Fluid Mech.,859, 691-730, 2019.
  33. N. Perakakis, A. Yazdani, G. E. Karniadakis, C. Mantzoros, “Omics, big data and machine learning as tools to propel understanding of biological mechanisms and to discover novel diagnostics and therapeutics.” Metabolism, 87:A1-A9, 2018.
  34. L. Bonfiglio, P. Perdikaris, J. del Aguila, G. E. Karniadakis, “A probabilistic framework for multidisciplinary design: Application to the hydrostructural optimization of supercavitating hydrofoils.” Int. J. Numer Methods Eng. 116:246-269, 2018.
  35. D. Zhang, H. Babaee, G. E. Karniadakis, “Stochastic domain decompostition via moment minimization.” SIAM J. Sci. Comput. 40(4), A2152-A2173, 2018.
  36. M. Raissi, P. Perdikaris, G. E. Karniadakis, “Numerical Gaussian processes for time-dependent and non-linear partial differential equations.SIAM J. Sci. Comput. 40(1), A172-182, 2018.
  37. M. Raissi, G. E. Karniadakis, “Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations.” J. Comp. Phys. 357(15), 125-141, 2018.
  38. Y. H. Tang, D. Zhang, G. E. Karniadakis, “An Atomistic Fingerprint Algorithm for Learning Ab Initio Molecular Force Field.” J. Chem. Phys. 148, 034101, 2018.
  39. L. Zhao, Z. Li, B. Caswell, J. Ouyang, G. E. Karniadakis, “Active learning of constitutive relation from mesoscopic dynamics for macroscopic modeling of non-Newtonian flows.” J Comp. Phys. 363, 116-127, 2018.
  40. L. Bonfiglio, P. Perdikaris, S. Brizzolara, G. E. Karniadakis, “Multi-fidelity optimization of super-cavitating hydrofoils.” Comput. Methods Appl. Mech. Engrg. 332, 63-85, 2018.
  41. M. Raissi, P. Perdikaris, G. E. Karniadakis, “Physics informed deep learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations.  arXiv:1711.10561v1.
  42. M. Raissi, P. Perdikaris, G. E. Karniadakis, “Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations.”
  43. M. Xu, D. P. Papageorgiou, S. Z. Abidi, M. Dao, H. Zhao, G. E. Karniadakis, “A deep convolutional neural network for classification of red blood cells in sickle cell anemia.” PLOS Comput. Biol. 13(10):e1005746, 2017.
  44. G. Pang, P. Perdikaris, W. Cai, G. E. Karniadakis, “Discovering variable fractional orders of advection–dispersion equations from field data using multi-fidelity Bayesian optimization.” J. Comput. Phys. 348:694-714, 2017.
  45. P. Perdikaris, M. Raissi, A. Damianou, N. D. Lawrence, G. E. Karniadakis, “Nonlinear information fusion algorithms for data-efficient multi-fidelity modeling. P. R. Soc. A. 473(2198), 2017.
  46. M. Raissi, P. Perdikaris, G. E. Karniadakis, “Inferring solutions of differential equations using noisy multi-fidelity data.” J. Comput. Phys. 335, 736-746, 2017.
  47. M. Raissi, P. Perdikaris, G. E. Karniadakis, “Machine learning of linear differential equations using Gaussian processes.” J. Comput. Phys. 348, 683-693, 2017.
  48. L. Bonfiglio, P. Perdikaris, S. Brizzolara, G. E. Karniadakis, “A multi-fidelity framework for investigating the performance of super-cavitating hydrofoils under uncertain flow conditions.” 19th AIAA Non-Deterministic Approaches Conference 1328, 2017.
  49. P. Prempraneerach, P. Perdikaris, G. E. Karniadakis, C. Chryssostomidis, “Sea Surface Temperature estimation from satellite observations and in-situ measurements using multifidelity Gaussian Process regression.” In Digital Arts, Media and Technology (ICDAMT), International Conference on (pp. 28-33). IEEE, 2017.
  50. S. Lee, I. G. Kevrekidis, G. E. Karniadakis, “A resilient and efficient CFD framework: Statistical learning tools for multi-fidelity and heterogeneous information fusion.” J. Comput. Phys. 344, 516-533, 2017.
  51. S. Lee, I. G. Kevrekidis, G. E. Karniadakis, “A general CFD framework for fault-resilient simulations based on multi-resolution information fusion.” J. Comput. Phys. 347, 290-304, 2017.
  52. L. Parussini, D. Venturi, P. Perdikaris, G. E. Karniadakis, “Multi-fidelity Gaussian process regression for prediction of random fields.” J. Comput. Phys. 336, 36-50, 2017.
  53. H. Babaee, P. Perdikaris, C. Chryssostomidis, G. E. Karniadakis, “Multi-fidelity modelling of mixed convection based on experimental correlations and numerical simulations.” J. Fluid Mech809:895-917, 2016.
  54. M. Raissi, G. E. Karniadakis, “Deep Multi-Fidelity Gaussian Processes.” arXiv preprint, 2016.
  55. P. Perdikaris, D. Venturi, G. E. Karniadakis, “Multifidelity information fusion algorithms for high-dimensional systems and massive data sets.” SIAM J. Sci. Comput38(4), B521-B538, 2016.
  56. P. Perdikaris, G. E. Karniadakis, “Model inversion via multi-fidelity Bayesian optimization: a new paradigm for parameter estimation in haemodynamics, and beyond.” J. Royal Soc. Interface. 13(118), 20151107, 2016.
  57. P. Perdikaris, D. Venturi, J.O Royset, G. E. Karniadakis, “Multi-fidelity modeling via recursive co-kringing and Gaussian Markov random fields.” P. R. Soc. A. 471, 20150018, 2015.
  58. S. Lee, I. G., Kevrekidis, G. E. Karniadakis, “Resilient algorithms for reconstructing and simulating gappy flow fields in CFD.” Fluid Dynamics Research47(5), 051402, 2015.
  59. A. Yakhot, T. Anor, G. E. Karniadakis, “A reconstruction method for gappy and noisy arterial flow data.” IEEE Transactions on Medical Imaging. 26(12):1681-97, 2007.
  60. H. Gunes, S. Sirisup, G. E. Karniadakis, “Gappy data: To Krig or not to Krig?” J. Comput. Phys. 212(1):358-82, 2006.